Zero-one Inflated Beta Distributions

The models were fitted using a zero-one inflated Beta distribution, which models the data as a Beta distribution for nLEDs excluding 0 and 1, and a Bernoulli distribution for binary nLEDs of 0 and 1. Thus, predictors in the model can affect four distributional parameters: \(\mu\) (mu), the mean of the nLEDs excluding 0 and 1; \(\phi\) (phi), the precision of the nLEDs excluding 0 and 1; \(\alpha\) (alpha; termed zoi - or zero-one inflation in brms) the probability of an nLED of 0 or 1; and \(\gamma\) (gamma; termed coi - or conditional-one inflation in brms), the conditional probability of a 1 given a 0 or 1 has been observed. Larger values for these parameters are associated with (a) higher mean nLEDs in the range exluding 0 and 1, (b) tighter distributions of the nLEDs in the range excluding 0 and 1 (i.e. less variance), (c) more zero-one inflation in nLEDs, and (d) more one-inflation relative to zero-inflation in nLEDs. Predictors in this model can influence any and all distributional parameters in the model at once. For these models, a logit link is used for the \(\mu\), \(\alpha\), and \(\gamma\) distributional parameters, and a log link is used for the \(\phi\) distributional parameter.

Model Fitting

Variety Exposure

Posterior means and credible intervals are provided in the table below.

We evaluated equivalence in the nLEDs between variety exposure conditions by determining a region of practical equivalence (ROPE) by which any effects between the reported bounds are deemed to be practically equivalent to 0. In all instances, this is determined at the 90% credible interval (CI) bound of the highest density interval (HDI), which is typically more stable than larger bounds (CITATION: Kruschke). We report the proportion of the HDI contained within the ROPE region along with bounds of this interval. Where HDIs are entirely contained by the equivalence bounds, equivalence is accepted. Where HDIs are entirely outside the equivalence bounds, equivalence is rejected. Uncertaintainty is assigned to any HDIs that cross the equialence bounds in either (or both) directions.

While nLEDs in the Variety Mismatch condition are generally lower than those in the Variety Match condition, 30% of the difference scores are contained by the equivalence bounds. Similarly, while nLEDs are generally higher in the two intervention conditions (Variety Mismatch Social and Dialect Literacy) when compared to the Variety Mismatch condition, around half of the difference scores are contained by the equivalence bounds. All other differences are largely undecided.

Word Type by Variety Exposure

We also looked at whether there are any differences in performance for different word types across conditions during the vocabulary testing phase.

Posterior means and credible intervals are provided in the table below.

In all instances there is some evidence that performance is better for non-contrastive words relative to contrastive words. However, equivalence between the two word types is only confidently rejected for the Dialect Literacy condition. Thus, it is likely that performance was worse for contrastive words relative to non-contrastive words in the Dialect Literacy condition in the Vocabulary Test.

Variety Exposure

Posterior means and credible intervals are provided in the table below.

While the ROPE is undecided, there does not seem to be any reliable differences across the Variety Exposure conditions in regards to overall performance.

Variety Exposure for Novel Words Only

Posterior means and credible intervals are provided in the table below.

A similar pattern appears for the novel words only as with all words.

Word Type by Variety Exposure

We also looked at whether there are any differences in performance for different word types across conditions during the testing phase.

There appears to be some differences in word types within groups, but are these differences reliable?

It seems that there is a clear effect of word type (for contrastive vs. non-contrastive words) for both tasks in the Dialect Literacy condition, and for reading only in the Variety Mismatch Social condition. Additionally, there is some evidence for an effect of word type in the Variety Mismatch condition, with only around 6% of the HDI in the the equivalence bounds. This suggests that performance is impaired for contrastive words only when participants are exposed to a dialect.

Posterior means and credible intervals are provided in the table below.

We can also directly compare the differences in performance for contrastive words relative to non-contrastive words.

These results reflect those in the plots above. Are any differences reported here reliable?

Equivalence is confidently rejected for the reading task for the two intervention conditions (Variety Mismatch Social and Dialect Literacy). Additionally, for the reading task the majority of the HDI for the Variety Mismatch condition is outside of the ROPE. For the spelling task, equivalence is rejected for the Dialect Literacy condition. However, all other contrasts show that most of the HDI is contained by the ROPE (suggesting equivalence).

Finally, we ask whether or not the contrastive effect is stronger in the Variety Mismatch Social condition relative to the Variety Mismatch condition.

Any differences here are slight, and are mainly contained by the ROPE in both tasks. This suggests that there are no substantial differences in the magnitude of the effect between contrastive and non-contrastive words across these two conditions.

Variety Exposure for Novel Words Only

It’s quite difficult to make out an overall pattern here, so instead we performed a median split based on the vocabulary test performance, and we categorised these into those with high and low nLEDs in the vocabulary test.

We focused specifically on novel words to see whether any differences occur for novel word decoding. The following analyses summarise the patterns in the covariate model.

It’s clear from the figure that performance is generally worse in the testing phase for those with high mean nLEDs during the vocabulary test. Are there any differences if we directly compare these difference scores?

Variety Match and Mismatch seem to differ for the spelling task, while all other contrasts seem to indicate equivalent performance across conditions. Is this borne out in the data?

Here, there is evidence that spelling performance between those with high and low nLEDs in the vocabulary test varies between the Variety Match and Variety Mismatch conditions. Viewing the first figure, it is clear that there is a larger discrepancy in performance between those with high and low nLEDs in the vocabulary test in the Variety Mismatch condition relative to the Variety Match condition. While there is a similar trend in the other two variety conditions (i.e. Variety Mismatch Social and Dialect Literacy), there is not enough evidence here to support the claim that the differences between those with high and low nLEDs in the vocabulary test vary substantially in the Variety Mismatch Social and Dialect Literacy conditions relative to the Variety Mismatch condition.

Next, we looked at whether or not contrastive words differed to non-contrastive words across the high and low nLED groups. First off, what do the nLEDs look like in these groups?

That’s quite a lot to parse in one go, so we can instead look at the difference scores between contrastive and non-contrastive words for this same comparison.

In the Variety Mismatch condition, the difference in the reading task between the two word types seems to stem from those with low nLEDs during the vocabulary test, while the difference is consistent across groups in both of the Dialect Intervention conditions. But, any effect in spelling is only found for those with high nLEDs in the vocabulary test following the Dialect Literacy training. Does the data back this up?

NOTE: The posterior samples for the covariate models is relatively low when using the median split. Increasing the number of samples may give us more stable estimates.